1. Kinetic Theory of Gases

2. Ideal GasThe number of molecules is largeThe average separation between molecules is largeMolecules moves randomlyMolecules obeys Newtons LawMolecules collide elastically with each other and with the wallConsists of identical molecules 3. The Ideal Gas Law PV = nRT in Kn: the number of moles in the ideal gasNtotal number n=NA of molecules Avogadros number: the number of atoms, molecules, etc, in a mole of a substance: NA=6.02 x 1023/mol.R: the Gas Constant: R = 8.31 J/mol K 4. Pressure and Temperature Pressure: Results from collisions of molecules on the surface ForceF Pressure: P=AAreadp Force:F= Rate of momentumdtgiven to the surfaceMomentum: momentum given by each collisiontimes the number of collisions in time dt 5. Only molecules moving toward the surface hitthe surface. Assuming the surface is normal tothe x axis, half the molecules of speed vx movetoward the surface.Only those close enough to the surface hit itin time dt, those within the distance vxdtThe number of collisions hitting an area A intime dt is1 N A vx dt2V Average densityThe momentum given by each collision tothe surface 2mvx 6. Momentum in time dt: 1 N dp = (2mv x ) A v x dt 2 VForce:dp 1 N F== (2mvx ) A v xdt 2V Pressure: F N 2 P = = mv xA VNot all molecules have the same v x average v2 x N 2P = mv x V 7. 2vx1 2 1 2 23 3 (2 = v = v x + v y + vz ) 21 2 1 2vx = v = vrms3 3 vrms is the root-mean-square speed222 vx + vy + vzvrms = v 2 =31N 2 2 N12Pressure: P =mv = mv3V3 V 2 Average Translational Kinetic Energy:1 2 1 2 K = mv = mvrms22 8. 2 NPressure:P = K3 V2FromPV = N K and PV = nRT3 3 nRT 3Temperature: K= = k BT 2 N 2R23Boltzmann constant: k B == 1.38 10 J/KNA 9. 12From PV = N mvrms 3 Nand PV = nRT =RT NA Avogadros numberN = nN A3RT vrms = M Molar mass M = mN A 10. Internal EnergyFor monatomic gas: the internal energy = sumof the kinetic energy of all molecules:33 Eint = N K = nN A k BT = nRT223Eint = nRT T2 11. Mean Free Path Molecules collide elastically with other molecules Mean Free Path : average distance between two consecutive collisions 1= 22d N / Vthe bigger the molecules the more molecules the more collisions the more collisions 12. Q = cm TMolar Specific HeatEint = Q W 3 Eint = nRT 2Definition:For constant volume: Q = nCV TFor constant pressure: Q = nC p TThe 1st Law of Thermodynamics: 3 Eint= nRT = Q W (Monatomic) 2 13. 3nRT = Q WConstant Volume 2(Monatomic) Q = nCV TW = PdV = 0 3 Eint = nRT32nRT = nCV T23CV = R2Eint = nCV T 14. 3nRT = Q W2Constant Pressure (Monatomic) Q = nC p TW = PV = nRT3nRT = nC p T nRT2 5CV = Cp R Cp = R 2Cp5 = =CV3 15. 1st LawdEint = dQ dWAdiabatic Process Ideal Gas LawpV = nRT (Q=0)Eint = nCV TdEint = dW = pdVC p = CV + R = nCV dTCp = pdV CVpdV + Vdp = nRdT = nR nCV Divide by pV:dV dp C p CV dVdV+ = = (1 )V p CV V V 16. dV dpdV Ideal Gas Law+ = (1 )pV = nRTV pVdpdV +=0 pVln p + ln V = ln( pV ) = const.pV = const.nRT ()V = const. V 1TV = const. 17. Equipartition of EnergyThe internal energy of non-monatomicmolecules includes also vibrational androtational energies besides thetranslational energy.Each degree of freedom has associated with1it an energy of k BT per molecules.2 18. Eint = nCV T Monatomic Gases3 translational degrees of freedom: 3 3 Eint = kBT nN A = nRT 2 2 1 dEint 3 CV = = R n dT2 19. Eint = nCV TDiatomic Gases3 translational degrees of freedom2 rotational degrees of freedom2 vibrational degrees of freedomHOWEVER, different DOFs require differenttemperatures to excite. At room temperature,only the first two kinds are excited: 55 Eint = nRT CV = R 22